# RATIONAL MECHANICS

## Learning outcomes of the course unit

By means of the frontal lessons, the student should be independently able to understand and to schematize by means of mathematical models simple phisical phenomena involving particles and rigid bodies. Moreover, the student should acquire the aptitude to learn, with critical mind, the theachings typical of the Mechanical Engineering degree.

## Prerequisites

There are no mandatory propedeuticities, but Mathematical Analysis, Geometry and Phisics I are strongly recommended.

## Course contents summary

Rational (or Theoretical) Mechanics is the science that studies the general laws of mechanical motions of material bodies (particles, rigid bodies) and establishes general procedures and methods for solving problems involving such motions. Being a mathematical science, Rational Mechanics deals not with real phisical objects but with abstract schematized conceptions. Consequently, Rational Mechanics il the link between the pure mathematics courses (Mathematical Analisys, Geometry) and the courses of engineering contents (expecially, Applied Mechanics).

It is customary to divide the course into three basic parts: Statics (the study of the rules of composition of forces and the conditions of equilibrium of rigid bodies), Kinematics (the study of motions of bodies from the geometric viewpoint alone, without reference to the forces acting on the bodies) and Dynamics (the study of the relationships between the motions of material bodies and the forces acting on them) . In particular, we emphasize the plane kinematics of the rigid bodies, and the dynamics of the one-dimensional mechanical systems.

## Course contents

Free vectors theory - geometry of the plane curves - kinematics of the element - kinematics of the rigid body - relative kinematics - rigid plane motiond - polar curves - statics and dynamics of the element - cardinal equations - applied vectors theory - one-dimensional mechanical systems (Lagrange's equations) - statics and dynamics of the rigid body - statics and dynamics of the articulate systems.

## Recommended readings

All the lectures and the exercises are available to students and shared on Elly platform. Moreover, the duplicate lectures notes provided by the lecturer are available on the Libreria Universitaria Santa Croce. In addition to this material, the student can personally study some of the topics discussed during the course in the following books: D.Graffi "Elementi di meccanica razionale" - P.G.Bordoni "Lezioni di meccanica razionale" - P.Biscari, T.Ruggeri, G.Saccomanni, M.Vianello "Meccanica razionale per l'ingegneria" - G.Frosali, E.Minguzzi "Meccanica Razionale per l'ingegneria" - P.Benvenuti, G.Maschio "Complementi ed esercizi di meccanica razionale".

## Teaching methods

The course counts 6 CFUs (one CFU , University Credits equals one ECTS credit and represent the workload of student during educational activities aimed at passing the exams), theat correspond to 48 hours of lectures. The didactit activities are composed of frontal lessons the significant part of which (almost half) will be dedicated to pratical work. This consists of mechanical examples and exercises that aim to illustrate and apply the concepts presented in the theoretical part of the course. The same type of exercises will be proposed in the first written examination.

## Assessment methods and criteria

The exam consists of twoo written tests and one possible oral test. The first written test (of 3 hours) contains one or more exercises. The students have to study the equilibrium position of a mechanical system (usually, two bodies) and to determine its motion equations. The utmost importance is given to the proceeding whereas the correctness of the numerical result it's not so important. The second written test (of 1.5 hours) will be held on 3/4 days after the first one. It consists of theoretical questions: definitions, properties, theorems, demonstrations,simple applications. A lot of examples of such written tests are available to students and shared on Elly platform.

The vote v(1),v(2) respectively of the first and of the second written test are in the range 0-30 , and the final vote v(f) is given by the formula: v(f) = (3/5)v(1) + (2/5)v(2) . If v(f) is below 18/30 the exam is no passed. If v(f) is at least 18/30 but v(1) or v(2) is below 16/30 , it is necessary to do also an oral test. In the other cases the exam is exceeded. When the exam is exceeded , the student can decide to do also an oral test.

The mark "cum laude" cam be acquired only after the oral test.